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Engineering Calculus Notes 51

# Engineering Calculus Notes 51 - b −→ c Figure 1.23...

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1.3. LINES IN SPACE 39 We will use these ideas to prove the following. Theorem 1.3.2. In any triangle, the three lines joining a vertex to the midpoint of the opposite side meet at a single point. Proof. Label the vertices of the triangle A , B and C , and their position vectors −→ a , −→ b and −→ c , respectively. Label the midpoint of each side with the name of the opposite vertex, primed; thus the midpoint of BC (the side opposite vertex A ) is A (see Figure 1.23 ). From Remark 1.3.1 we see x y z b b b A A B B C C A B C −→ a −→
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Unformatted text preview: b −→ c Figure 1.23: Theorem 1.3.2 that the position vectors of the midpoints of the sides are −−→ O A ′ = 1 2 ( −→ b + −→ c ) −−→ O B ′ = 1 2 ( −→ c + −→ a ) −−→ O C ′ = 1 2 ( −→ a + −→ b ) , and so the line ℓ A through A and A ′ can be parametrized (using r as the parameter) by −→ p A ( r ) = (1 − r ) −→ a + r 2 ( −→ b + −→ c ) = (1 − r ) −→ a + r 2 −→ b + r 2 −→ c ....
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